Appendix 1: The derivation of Proposition 3.1
$$ \begin{aligned} V_{L} (t) = & E_{t} [ - L(t) + {\frac{1}{{{ \exp }\{ \varphi (t,t)*h\} }}}[L(t)*{ \exp }\{ c(t)*h\} - L(t + h)] \\ & + {\frac{1}{{{ \exp }\{ [\varphi (t,t) + \varphi (t + h,t + h)]*h\} }}}[L(t + h)*{ \exp }\{ c(t + h)*h\} - L(t + 2h)] \\ & + {\frac{1}{{{ \exp }\{ [\varphi (t,t) + \varphi (t + h,t + h) + \varphi (t + 2h,t + 2h)]*h\} }}}[L(t + 2h)*{ \exp }\{ c(t + 2h)*h\} - L(t + 3h)] \\ & + \cdots + \cdots + \cdots + {\frac{1}{{\exp \{ \sum\limits_{{i = {\frac{t}{h}}}}^{{{\frac{T}{h}} - 2}} {\varphi (ih,ih)*h} \} }}}[L(T - 2h)*\exp \{ c(T - 2h)*h\} - L(T - h)] \\ & + {\frac{1}{{{ \exp }\{ \sum\limits_{{i = {\frac{t}{h}}}}^{{{\frac{T}{h}} - 1}} {\varphi (ih,ih)*h} \} }}}[L(T - h)*{ \exp }\{ c(T - h)*h\} ]] \\ = & E_{t} [ - L(t) + {\frac{J(t)}{J(t + h)}}[L(t)*{ \exp }\{ c(t)*h\} - L(t + h)] + {\frac{J(t)}{J(t + 2h)}}[L(t + h)*{ \exp }\{ c(t + h)*h\} - L(t + 2h)] \\ & + {\frac{J(t)}{J(t + 3h)}}[L(t + 2h)*{ \exp }\{ c(t + 2h)*h\} - L(t + 3h)] + \cdots \\ & + {\frac{J(t)}{J(T - h)}}[L(T - 2h)*{ \exp }\{ c(T - 2h)*h\} - L(T - h)] + {\frac{J(t)}{J(T)}}[L(T - h)*{ \exp }\{ c(T - h)*h\} ]] \\ = & E_{t} [ - L(t) \\ & + [{\frac{J(t)}{J(t + h)}}*L(t)*{ \exp }\{ c(t)*h\} + {\frac{J(t)}{J(t + 2h)}}*L(t + h)*{ \exp }\{ c(t + h)*h\} + {\frac{J(t)}{J(t + 3h)}}*L(t + 2h)*{ \exp }\{ c(t + 2h)*h\} \\ & + \cdots + {\frac{J(t)}{J(T - h)}}*L(T - 2h)*{ \exp }\{ c(T - 2h)*h\} + {\frac{J(t)}{J(T)}}*L(T - h)*{ \exp }\{ c(T - h)*h\} ] \\ & - [{\frac{J(t)}{J(t + h)}}*L(t + h) + {\frac{J(t)}{J(t + 2h)}}*L(t + 2h) + {\frac{J(t)}{J(t + 3h)}}*L(t + 3h) + \cdots + {\frac{J(t)}{J(T - h)}}*L(T - h)]] \\ = & E_{t} [ - L(t) + J(t)[\sum\limits_{k = t/h}^{T/h - 1} {{\frac{{L(kh)*{ \exp }\{ c(kh)*h\} }}{J(kh + h)}}} - \sum\limits_{k = t/h}^{T/h - 2} {{\frac{L(kh + h)}{J(kh + h)}}} ]] \\ = & E_{t} [ - L(t) + J(t)[\sum\limits_{k = t/h}^{T/h - 1} {{\frac{{L(kh)*{ \exp }\{ c(kh)*h\} - L(kh + h)}}{J(kh + h)}} + } {\frac{L(T)}{J(T)}}]] \\ = & E_{t} [ - L(t) + J(t)*\sum\limits_{k = t/h}^{T/h - 1} {{\frac{{L(kh)*{ \exp }\{ c(kh)*h\} - L(kh + h)}}{J(kh + h)}} + } {\frac{J(t)*L(T)}{J(T)}}]. \quad {\text{Q.E.D.}}\\ \end{aligned} $$
Appendix 2: The derivation of Lemma 4.1
From Eq. 4, the net present value of the credit card loan can be expressed as:
$$ V_{L} (t) = E_{t} \left[ { - L(t) + \sum\limits_{i = t/h}^{T/h - 2} {{\frac{{L(ih)\exp \left( {c(ih)h} \right) - L((i + 1)h)}}{{\exp \left( {\sum\limits_{j = t/h}^{i} {\varphi (jh,jh)h} } \right)}}}} + {\frac{{L(T - h)\exp \left( {c(T - h)h} \right)}}{{\exp \left( {\sum\limits_{j = t/h}^{T/h - 1} {\varphi (jh,jh)h} } \right)}}}} \right] $$
$$ = E_{t} \left[ {\sum\limits_{i = t/h}^{T/h - 2} {{\frac{{L(ih)\exp \left( {c(ih)h} \right)}}{{\exp \left( {\sum\limits_{j = t/h}^{i} {\varphi (jh,jh)h} } \right)}}}} + {\frac{{L(T - h)\exp \left( {c(T - h)h} \right)}}{{\exp \left( {\sum\limits_{j = t/h}^{T/h - 1} {\varphi (jh,jh)h} } \right)}}}} \right.\;\; $$
$$ \left. {\; - {\frac{{L(t)\exp \left( {\varphi (t,t)h} \right)}}{{\exp \left( {\varphi (t,t)h} \right)}}} - \sum\limits_{i = t/h}^{T/h - 2} {{\frac{{L((i + 1)h)\exp \left( {\varphi \left( {(i + 1)h,(i + 1)h} \right)h} \right)}}{{\exp \left( {\sum\limits_{j = t/h}^{i} {\varphi (jh,jh)h} } \right)\exp \left( {\varphi \left( {(i + 1)h,(i + 1)h} \right)h} \right)}}}} } \right] $$
$$ = E_{t} \left[ {\sum\limits_{i = t/h}^{T/h - 1} {{\frac{{L(ih)\left[ {\exp \left( {c(ih)h} \right) - \exp \left( {\varphi (ih,ih)h} \right)} \right]}}{{\exp \left( {\sum\limits_{j = t/h}^{i} {\varphi (jh,jh)h} } \right)}}}} } \right]. \quad {\text{Q.E.D.}}$$
Appendix 3: The derivation of Proposition 4.2
From Eq. 25, we have:
$$ r(t) = f(0,t) + \sigma_{r}^{2} (e^{{ - a_{r} t}} - 1)^{2} /(2a_{r}^{2} ) + \int_{0}^{t} {\sigma_{r} e^{{ - a_{r} (t - u)}} {\text{d}}\widetilde{{W_{r} }}(u)} , $$
(33)
We can then write:
$$ \int_{0}^{t} {r(u){\text{d}}u} = \int_{0}^{t} {f(0,u){\text{d}}u} + \int_{0}^{t} {\sigma_{r}^{2} (e^{{ - a_{r} u}} - 1)^{2} /(2a_{r}^{2} )} {\text{d}}u + \int_{0}^{t} {\int_{0}^{v} {\sigma_{r} e^{{ - a_{r} (v - u)}} {\text{d}}\widetilde{{W_{r} }}(u)} } {\text{d}}v $$
$$ = \int_{0}^{t} {f(0,u){\text{d}}u} + \int_{0}^{t} {\sigma_{r}^{2} (e^{{ - a_{r} u}} - 1)^{2} /(2a_{r}^{2} )} {\text{d}}u + \int_{0}^{t} {\int_{u}^{t} {\sigma_{r} e^{{ - a_{r} (v - u)}} {\text{d}}v{\text{d}}\widetilde{{W_{r} }}(u)} } $$
$$ = \int_{0}^{t} {f(0,u){\text{d}}u} + \int_{0}^{t} {\sigma_{r}^{2} (e^{{ - a_{r} u}} - 1)^{2} /(2a_{r}^{2} )} {\text{d}}u + \int_{0}^{t} {(\sigma_{r} /a_{r} )\left[ {1 - e^{{ - a_{r} (t - u)}} } \right]} {\text{d}}\widetilde{{W_{r} }}(u). $$
(34)
In a similar way, we can restate Eq. 26 as:
$$ s(t) = s(0,t) + \sigma_{s}^{2} (e^{{ - a_{s} t}} - 1)^{2} /(2a_{s}^{2} ) + \int_{0}^{t} {\sigma_{s} e^{{ - a_{s} (t - u)}} {\text{d}}\widetilde{{W_{s} }}(u)} , $$
(35)
Hence we obtain
$$ \int_{0}^{t} {s(u){\text{d}}u} = \int_{0}^{t} {f(0,u){\text{d}}u} + \int_{0}^{t} {\sigma_{s}^{2} (e^{{ - a_{s} u}} - 1)^{2} /(2a_{s}^{2} )} {\text{d}}u + \int_{0}^{t} {(\sigma_{s} /a_{s} )\left[ {1 - e^{{ - a_{s} (t - u)}} } \right]} {\text{d}}\widetilde{{W_{s} }}(u). $$
(36)
Equations 33–36 are four normal random variables with the following respective means, variances and covariances.
$$ \mu_{1} (t) \equiv E\left( {\int_{0}^{t} {r(u){\text{d}}u} } \right) = \int_{0}^{t} {f(0,u){\text{d}}u} + \int_{0}^{t} {\left[ {\sigma_{r}^{2} \left( {\exp \left( { - a_{r} u} \right) - 1} \right)^{2} /(2a_{r}^{2} )} \right]} {\text{d}}u $$
$$ \mu_{2} (t) \equiv E\left( {r(t)} \right) = f(0,t) + \sigma_{r}^{2} \left( {\exp \left( { - a_{r} t} \right) - 1} \right)^{2} /(2a_{r}^{2} ) $$
$$ \mu_{3} (t) \equiv E\left( {\int_{0}^{t} {s(u){\text{d}}u} } \right) = \int_{0}^{t} {s(0,u){\text{d}}u} + \int_{0}^{t} {\left[ {\sigma_{s}^{2} \left( {\exp \left( { - a_{s} u} \right) - 1} \right)^{2} /(2a_{s}^{2} )} \right]} {\text{d}}u $$
$$ \mu_{4} (t) \equiv E\left( {s(t)} \right) = s(0,t) + \sigma_{s}^{2} \left( {\exp \left( { - a_{s} t} \right) - 1} \right)^{2} /(2a_{s}^{2} ) $$
$$ \sigma_{1}^{2} (t) \equiv Var\left( {\int_{0}^{t} {r(u){\text{d}}u} } \right) $$
$$ = Var\left( {\int_{0}^{t} {(\sigma_{r} /a_{r} )\left[ {1 - \exp \left( { - a_{r} (t - u)} \right)} \right]} d\widetilde{{W_{r} }}(u)} \right) $$
$$ = \int_{0}^{t} {\left[ {\sigma_{r}^{2} \left( {1 - \exp \left( { - a_{r} (t - u)} \right)} \right)^{2} /a_{r}^{2} } \right]} {\text{d}}u $$
$$ \sigma_{2}^{2} (t) \equiv Var\left( {r(t)} \right) $$
$$ = Var\left( {\int_{0}^{t} {\sigma_{r} \left( { - a_{r} (t - u)} \right)} {\text{d}}\widetilde{{W_{r} }}(u)} \right) $$
$$ = \int_{0}^{t} {\sigma_{r}^{2} \exp \left( { - 2a_{r} (t - u)} \right)} {\text{d}}u $$
$$ \sigma_{3}^{2} (t) \equiv Var\left( {\int_{0}^{t} {s(u){\text{d}}u} } \right) $$
$$ = Var\left( {\int_{0}^{t} {(\sigma_{s} /a_{s} )\left[ {1 - \exp \left( { - a_{s} (t - u)} \right)} \right]} {\text{d}}\widetilde{{W_{s} }}(u)} \right) $$
$$ = \int_{0}^{t} {\left[ {\sigma_{s}^{2} \left( {1 - \exp \left( { - a_{s} (t - u)} \right)} \right)^{2} /a_{s}^{2} } \right]} {\text{d}}u $$
$$ \sigma_{4}^{2} (t) \equiv Var\left( {s(t)} \right) $$
$$ = Var\left( {\int_{0}^{t} {\sigma_{s} \left( { - a_{s} (t - u)} \right)} {\text{d}}\widetilde{{W_{s} }}(u)} \right) $$
$$ = \int_{0}^{t} {\sigma_{s}^{2} \exp \left( { - 2a_{s} (t - u)} \right)} {\text{d}}u $$
$$ \sigma_{12} (t) \equiv \text{cov} \left( {\int_{0}^{t} {r(u){\text{d}}u,\;\;r(t)} } \right) $$
$$ = \text{cov} \left( {\int_{0}^{t} {(\sigma_{r} /a_{r} )\left[ {1 - \exp \left( { - a_{r} (t - u)} \right)} \right]} {\text{d}}\widetilde{{W_{r} }}(u),\;\;\int_{0}^{t} {\sigma_{r} \exp \left( { - a_{r} (t - u)} \right){\text{d}}\widetilde{{W_{r} }}(u)} } \right) $$
$$ = \int_{0}^{t} {(\sigma_{r}^{2} /a_{r} )\exp \left( { - a_{r} (t - u)} \right)\left[ {1 - \exp \left( { - a_{r} (t - u)} \right)} \right]} {\text{d}}u $$
$$ = \left( {\sigma_{r}^{2} /(2a_{r}^{2} )} \right)\left( {1 - \exp \left( { - a_{r} t} \right)^{2} } \right) $$
$$ \sigma_{13} (t) \equiv \text{cov} \left( {\int_{0}^{t} {r(u){\text{d}}u,\;\;\int_{0}^{t} {s(u){\text{d}}u} } } \right) $$
$$ = \text{cov} \left( {\int_{0}^{t} {{\frac{{\sigma_{r} }}{{a_{r} }}}\left[ {1 - \exp \left( { - a_{r} (t - u)} \right)} \right]} {\text{d}}\widetilde{{W_{r} }}(u),\;\;\int_{0}^{t} {{\frac{{\sigma_{s} }}{{a_{s} }}}\left[ {1 - \exp \left( { - a_{s} (t - u)} \right)} \right]} {\text{d}}\widetilde{{W_{s} }}(u)} \right) $$
$$ = \int_{0}^{t} {(\sigma_{r} \sigma_{s} \rho )\left[ {1 - \exp \left( { - a_{r} (t - u)} \right)} \right]} \left[ {1 - \exp \left( { - a_{s} (t - u)} \right)} \right]/(a_{r} a_{s} ){\text{d}}u $$
$$ \sigma_{14} (t) \equiv \text{cov} \left( {\int_{0}^{t} {r(u){\text{d}}u,\;\;s(t)} } \right) $$
$$ = \text{cov} \left( {\int_{0}^{t} {(\sigma_{r} /a_{r} )\left[ {1 - \exp \left( { - a_{r} (t - u)} \right)} \right]} {\text{d}}\widetilde{{W_{r} }}(u),\;\;\int_{0}^{t} {\sigma_{s} \exp \left( { - a_{s} (t - u)} \right)} {\text{d}}\widetilde{{W_{s} }}(u)} \right) $$
$$ = \int_{0}^{t} {(\sigma_{r} \sigma_{s} \rho )\left[ {1 - \exp \left( { - a_{r} (t - u)} \right)} \right]} \exp \left( { - a_{s} (t - u)} \right)/a_{r} {\text{d}}u $$
$$ \sigma_{23} (t) \equiv \text{cov} \left( {r(t),\;\;\int_{0}^{t} {s(u){\text{d}}u} } \right) $$
$$ = \text{cov} \left( {\int_{0}^{t} {\sigma_{r} \exp \left( { - a_{r} (t - u)} \right)} {\text{d}}\widetilde{{W_{r} }}(u),\;\;\int_{0}^{t} {(\sigma_{s} /a_{s} )\left[ {1 - \exp \left( { - a_{s} (t - u)} \right)} \right]} {\text{d}}\widetilde{{W_{s} }}(u)} \right) $$
$$ = \int_{0}^{t} {(\sigma_{r} \sigma_{s} \rho )\left[ {1 - \exp \left( { - a_{s} (t - u)} \right)} \right]} \exp \left( { - a_{r} (t - u)} \right)/a_{s} {\text{d}}u $$
$$ \sigma_{24} (t) \equiv \text{cov} \left( {r(t),\;s(t)} \right) $$
$$ = \text{cov} \left( {\int_{0}^{t} {\sigma_{r} \exp \left( { - a_{r} (t - u)} \right)} {\text{d}}\widetilde{{W_{r} }}(u),\;\;\int_{0}^{t} {\sigma_{s} \exp \left( { - a_{s} (t - u)} \right)} {\text{d}}\widetilde{{W_{s} }}(u)} \right) $$
$$ = \int_{0}^{t} {(\sigma_{r} \sigma_{s} \rho )} \exp \left( { - a_{r} (t - u) - a_{s} (t - u)} \right){\text{d}}u $$
$$ \sigma_{34} (t) \equiv \text{cov} \left( {\int_{0}^{t} {s(u){\text{d}}u} ,\;\;s(t)} \right) $$
$$ = \text{cov} \left( {\int_{0}^{t} {(\sigma_{s} /a_{s} )\left[ {1 - \exp \left( { - a_{s} (t - u)} \right)} \right]} {\text{d}}\widetilde{{W_{s} }}(u),\;\;\int_{0}^{t} {\sigma_{s} \exp \left( { - a_{s} (t - u)} \right){\text{d}}\widetilde{{W_{s} }}(u)} } \right) $$
$$ = \int_{0}^{t} {(\sigma_{s}^{2} /a_{s} )\exp \left( { - a_{s} (t - u)} \right)\left[ {1 - \exp \left( { - a_{s} (t - u)} \right)} \right]} {\text{d}}u $$
$$ = \left( {\sigma_{s}^{2} /(2a_{s}^{2} )} \right)\left( {1 - \exp \left( { - a_{s} t} \right)^{2} } \right) \quad{\text{Q.E.D.}}$$
Appendix 4: The derivation of Corollary 4.3
If we do not take default risk into consideration, our closed-form solution will reduce to those of Jarrow and van Deventer (1998). This can be obtained by setting the parameters of default risk related parameters as zero. We show the results as follows:
Let \( a_{s} = \sigma_{s} = \rho = s(t) = s(0,t) = 0 \), then the closed form solution will be reduced to
$$ V_{L} (0) = L(0)\exp \left( {c(0) - \left( {\alpha_{3} + \beta_{2} } \right)r(0)} \right)\int_{0}^{\tau } {\exp \left( {(\alpha_{0} + \beta_{0} )t + \alpha_{1} t^{2} /2} \right)}\,\cdot\,M\left( {t,\alpha_{2} + \beta_{1} - 1,\alpha_{3} + \beta_{2} } \right){\text{d}}t - L(0)\exp \left( { - \alpha_{3} r(0)} \right)\int_{0}^{\tau } {\exp \left( {\alpha_{0} t + \alpha_{1} t^{2} /2} \right)}\,\cdot\,M\left( {t,\alpha_{2} - 1,\alpha_{3} + 1} \right){\text{d}}t. $$
where \( M(t,\gamma_{1} ,\gamma_{2} ) = E_{0} \left( {e^{{\gamma_{1} x_{1} + \gamma_{2} x_{2} }} } \right) \)
$$ \; = \;\exp \left( {\mu_{1} \gamma_{1} + \mu_{2} \gamma_{2} + \left[ {\sigma_{1}^{2} \gamma_{1}^{2} + 2\sigma_{12} \gamma_{1} \gamma_{2} + \sigma_{2}^{2} \gamma_{2}^{2} } \right]/2} \right) $$
$$ \mu_{1} (t) \equiv \int_{0}^{t} {f(0,u){\text{d}}u} + \int_{0}^{t} {\left[ {\sigma_{r}^{2} \left( {\exp \left( { - a_{r} u} \right) - 1} \right)^{2} /(2a_{r}^{2} )} \right]} {\text{d}}u $$
$$ \mu_{2} (t) \equiv f(0,t) + \sigma_{r}^{2} \left( {\exp \left( { - a_{r} t} \right) - 1} \right)^{2} /(2a_{r}^{2} ) $$
$$ \sigma_{1}^{2} (t) \equiv \int_{0}^{t} {\left[ {\sigma_{r}^{2} \left( {1 - \exp \left( { - a_{r} (t - u)} \right)} \right)^{2} /a_{r}^{2} } \right]} {\text{d}}u $$
$$ \sigma_{2}^{2} (t) \equiv \int_{0}^{t} {\sigma_{r}^{2} \exp \left( { - 2a_{r} (t - u)} \right){\text{d}}u} $$
$$ \sigma_{12} (t) \equiv \left( {\sigma_{r}^{2} /(2a_{r}^{2} )} \right)\left( {1 - \exp \left( { - a_{r} t} \right)^{2} } \right) $$
The above pricing formula is exact the same as that of Jarrow and van Deventer (1998). Q.E.D.
